On local motion of a general compressible viscous heat conducting fluid bounded by a free surface

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1 ANNALE POLONICI MAHEMAICI LIX.2 (1994 On local moion of a general compressible viscous hea conducing fluid bounded by a free surface by Ewa Zadrzyńska ( Lódź and Wojciech M. Zaja czkowski (Warszawa Absrac. he moion of a viscous compressible hea conducing fluid in a domain in R 3 bounded by a free surface is considered. We prove local exisence and uniqueness of soluions in obolev lobodeskiĭ spaces in wo cases: wih surface ension and wihou i. 1. Inroducion. In his paper we consider he moion of a viscous hea conducing fluid in a bounded domain Ω R 3 wih a free boundary. Le v = v(x, be he velociy of he fluid (i.e. v = (v 1, v 2, v 3, ϱ = ϱ(x, he densiy, ϑ = ϑ(x, he emperaure, f = f(x, he exernal force field per uni mass, r = r(x, he efficiency of hea sources per uni mass, p = p(ϱ, ϑ he pressure, µ and ν he consan viscosiy coefficiens, σ he consan coefficien of surface ension, κ he consan coefficien of hea conduciviy, c v = c v (ϱ, ϑ he specific hea a consan volume, p he exernal (consan pressure. hen he problem is described by he following sysem (see [4], Chs. 2 and 5: ϱ(v + v v + p µ v ν div v = ϱf in Ω, ϱ + div(ϱv = in Ω, (1.1 ϱc v (ϑ + v ϑ + ϑp ϑ div v κ ϑ µ 3 (v i,xj + v j,xi 2 (ν µ(div v 2 = ϱr in 2 Ω, i,j=1 n σhn = p n on, 1991 Mahemaics ubjec Classificaion: 35A5, 35R35, 76N1. Key words and phrases: free boundary, compressible viscuous hea conducing fluid, local exisence, anisoropic obolev spaces. Research suppored by KBN gran no

2 134 E. Zadrzyńska and W. M. Zaj aczkowski (1.1 [con.] v n = φ φ on, ϑ n = ϑ on, v = = v in Ω, ϱ = = ϱ in Ω, ϑ = = ϑ in Ω, where φ(x, = describes, n is he uni ouward vecor normal o he boundary (i.e. n = φ/ φ, Ω = (, Ω {}, Ω is he domain of he drop a ime and Ω = Ω is is iniial domain, = (, {}. Finally, = (v, p denoes he sress ensor of he form (1.2 = { ij } = { pδ ij + µ(v i,xj + v j,xi + (ν µδ ij div v} { pδ ij + D ij (v}, where i, j = 1, 2, 3, D = D(v = {D ij } is he deformaion ensor. Moreover, hermodynamic consideraions imply ha c v >, κ >, ν 1 3 µ >. By H we denoe he double mean curvaure of which is negaive for convex domains and can be expressed in he form (1.3 Hn = (x, x = (x 1, x 2, x 3, where ( is he Laplace Belrami operaor on. Le be given by x = x(s 1, s 2,, (s 1, s 2 U R 2, where U is an open se. hen (1.4 ( = g 1/2 g 1/2 ĝ αβ s α s β = g 1/2 g 1/2 g αβ (α, β = 1, 2, s α s β where he summaion convenion over repeaed indices is assumed, g = de{g αβ } α,β=1,2, g αβ = x α x β (x α = x/ s α, {g αβ } is he inverse marix o {g αβ } and {ĝ αβ } is he marix of algebraic complemens of {g αβ }. Le he domain Ω be given. hen by (1.1 5, Ω = {x R 3 : x = x(ξ,, ξ Ω}, where x = x(ξ, is he soluion of he Cauchy problem x (1.5 = v(x,, x = = ξ Ω, ξ = (ξ 1, ξ 2, ξ 3. herefore he ransformaion x = x(ξ, connecs Eulerian x and Lagrangian ξ coordinaes of he same fluid paricle. Hence (1.6 x = ξ + u(ξ, s ds X u (ξ,,

3 Moion of a hea conducing fluid 135 where u(ξ, = v(x u (ξ,,. Moreover, he kinemaic boundary condiion (1.1 5 implies ha he boundary is a maerial surface. hus, if ξ = hen X u (ξ, and = {x : x = X u (ξ,, ξ }. By he equaion of coninuiy (1.1 2 and (1.1 5 he oal mass M of he drop is conserved and he following relaion beween ϱ and Ω holds: (1.7 Ω ϱ(x, dx = M. he aim of his paper is o prove he local-in-ime exisence and uniqueness of soluions o problem (1.1 in obolev lobodeskiĭ spaces (see definiion in ec. 2. In he case of compressible baroropic fluid he corresponding drop problem has been considered by W. M. Zaj aczkowski in [13] and [16], while papers [14] and [15] refer o he global exisence of soluion o he same drop problem. Local exisence of soluions in he compressible baroropic case was also considered in [5], [6], [12], while in he incompressible baroropic case local exisence is proved in [2] and [1]. his paper consiss of four secions. In ecion 2 noaion and auxiliary resuls are presened. In ecion 3 we prove he local exisence and uniqueness of soluion o problem (1.1 in he case σ =. In his case here is no surface ension. Finally, ecion 4 concerns he local exisence and uniqueness of soluion o problem (1.1 in he case σ, i.e. when he shape of he free boundary of Ω is governed by surface ension. 2. Noaions and auxiliary resuls. In ecions 3 and 4 of his paper we use he anisoropic obolev lobodeskiĭ spaces W l,l/2 2 (Q, l R 1 + (see [3], of funcions defined in Q where Q = Ω Ω (, (Ω R 3 is a domain, or Q = (,, = Ω. We define W l,l/2 2 (Ω as he space of funcions u such ha [ u l,l/2 W 2 (Ω = (2.1 Dξ α u i 2 L 2 (Ω + ( α +2i=[l] + Ω [ α +2i [l] α +2i [l] Ω Ω D α ξ i u(ξ, D α ξ i u(ξ, 2 ξ ξ 3+2(l [l] dξ dξ d D α ξ i u(ξ, D α ξ i u(ξ, 2 1+2(l/2 [l/2] d d dξ D α ξ i u 2 2,Ω + α +2i=[l] + [D α ξ i u] 2 l/2 [l/2],2,ω, ] 1/2 <, ([D α ξ i u] 2 l [l],2,ω,ξ ] 1/2

4 136 E. Zadrzyńska and W. M. Zaj aczkowski where we use generalized (obolev derivaives, D α ξ = α 1 ξ 1 α 2 ξ 2 α 3 ξ 3, α j ξ j = α j / ξ α j j (j = 1, 2, 3, α = (α 1, α 2, α 3 is a muliindex, α = α 1 + α 2 + α 3, i = i / i and [l] is he ineger par of l. In he case when l is an ineger he second erms in he above formulae mus be omied, while in he case of l/2 being ineger he las erms in he above formulae mus be omied as well. imilarly o W l,l/2 2 (Ω, using local mappings and a pariion of uniy we inroduce he normed space W l,l/2 2 ( of funcions defined on = (,, where = Ω. We also use he usual obolev spaces W2(Q, l where l R +, Q = Ω (Ω R 3 is a bounded domain or Q =. In he case Q = Ω he norm in W2(Ω l is defined as follows: ( u W l 2 (Ω = α [l] + α =[l] ( α [l] D α ξ u 2 L 2 (Ω Ω Ω Dξ αu(ξ Dα ξ u(ξ 2 1/2 dξ dξ ξ ξ 3+2(l [l] D α ξ u 2 2,Ω + [D α ξ u] 2 l [l],2,ω 1/2, where he las erm is omied when l is an ineger. imilarly, by using local mappings and a pariion of uniy we define W l 2(. o simplify noaion we wrie u l,q = u l,l/2 W 2 (Q if Q = Ω or Q =, l ; u l,q = u W l 2 (Q if Q = Ω or Q =, l, and W, 2 (Q = W 2 (Q = L 2 (Q. Moreover, u Lp (Q = u p,q, 1 p. Nex inroduce he space Γ l,l/2 (Ω wih he norm u l,l/2 Γ (Ω = Dξ α u i,ω u l,,ω α +2i l and he space L p (, ; Γ l,l/2 (Ω wih he norm u Lp (, ;Γ l,l/2 (Ω u l,,p,ω, where 1 p. Moreover, le C α (Ω (α (, 1 denoe he Hölder space wih he norm u(ξ, u(ξ, u C α (Ω = sup Ω ( ξ ξ ; α

5 Moion of a hea conducing fluid 137 le C B (Ω be he space of coninuous bounded funcions on Ω wih he norm u C B (Ω = sup u(ξ, and le C 2,1 (Q (Q R 3 (, denoe he space of funcions u such ha Dξ α i u C (Q for α + 2i 2. Finally, he following seminorms are used: u κ,q = ( Ω u 2 2,Q 2κ 1/2 d, where Q = Ω (Ω R 3 is a bounded domain or Q =, and κ (, 1; [u] l,2,q = [u] l,2,q,ξ + [u] l,2,q,, where [u] l,2,q,ξ = [Dξ α u] i l [l],2,q,ξ, [u] l,2,q, = α +2i=[l] α +2i=[l] [D α ξ i u] l/2 [l/2],2,q,, Q = Ω J (Ω R 3 is a domain, J = (, or J = (, or Q = J. In he case when J = (, he seminorms [Dξ α i u] l [l],2,q,ξ and [Dξ α i u] l/2 [l/2],2,q, are defined in (2.1. In he case when J = (, we define he above seminorms in he same way. Le X be whichever of he funcion spaces menioned above. We say ha a vecor-valued funcion u = (u 1,..., u ν belongs o X if u i X for any 1 i ν. In he sequel we shall use various noaions for derivaives of u (where u is a scalar- or vecor-valued funcion u = (u 1, u 2, u 3. If u is a scalar-valued funcion we denoe by Dξ ku (where ξ Ω R3 he vecor of all derivaives of u of order k, i.e. Dξ ku = (Dα ξ u α =k. imilarly, if u = (u 1, u 2, u 3 we denoe by Dξ ku he vecor (Dα ξ u j α =k, j=1,2,3. By Dξ, k u we denoe he vecor (Dξ α i u j α +2i=k,j=1,2,3 in he case when u = (u 1, u 2, u 3 and he vecor (Dξ α i u α +2i=k in he scalar case. Hence Dξ k u = Dξ α u and Dξ,u k = Dξ α u i. α =k α +2i=k We also use he noaion ξ u D 1 ξ u or u ξ D 1 ξ u. Nex, we denoe by u v eiher he scalar produc of vecors u and v, or he produc of marices u and v.

6 138 E. Zadrzyńska and W. M. Zaj aczkowski Finally, we denoe by Dξ, k udl ξ,v he following number: (2.2 Dξ,uD k ξ,v l = p s Dξ α u i m D β ξ j v n, α +2i=k β +2j=l m=1 n=1 where u = (u 1,..., u p, v = (v 1,..., v s (k, l, p > 1, s > 1. he produc of more han wo such facors is defined similarly. We use he following lemmas. Lemma 2.1. he following imbedding holds: (2.3 W l r(ω L α p (Ω (Ω R 3, where α + 3/r 3 l, l Z, 1 p, r ; L α p (Ω is he space of funcions u such ha D α ξ u p,ω < ; W l r(ω is he obolev space. Moreover, he following inerpolaion inequaliies hold: (2.4 D α ξ u p,ω cε 1 κ D l ξu r,ω + cε κ u r,ω, where κ = α /l + 3/(lr 3/(lp < 1, ε is a parameer and c > is a consan independen of u and ε; (2.5 D α ξ u q, cε 1 κ D l ξu r,ω + cε κ u r,ω, where κ = α /l + 3/(lr 2/(lq < 1, ε is a parameer and c > is a consan independen of u and ε. Lemma 2.1 follows from heorem 1.2 of [3]. Lemma 2.2 (see [7]. For sufficienly regular u we have i u( 2l 1 2i,Ω c( u 2l,Ω + i u( 2l 1 2i,Ω, where 2i 2l 1, l N and c > is a consan independen of. Lemma 2.3. Le u(ξ, = for. hen d d u(ξ, u(ξ, 2 2,Q 1+2α + 1 α d d u(ξ, u(ξ, 2 2,Q 1+2α u( 2 2,Q 2α d, where Q = Ω (Ω R 3 is a domain or Q = = Ω, and α R. (2.6 Lemma 2.4. Le τ (, 1. hen for all u W,τ/2 2 (Ω, u 2 2,Ω d τ c 1 d d u(, u(, 2 2,Ω 1+τ + c 2 τ u 2 2,Ω d,

7 Moion of a hea conducing fluid 139 where c 1, c 2 do no depend on and u. For = he las erm in (2.6 vanishes. his was shown in [11], Lemma Local exisence in he case σ =. In order o prove local exisence of soluions of (1.1 we rewrie i in he Lagrangian coordinaes inroduced by (1.5 and (1.6: ηu µ 2 uu ν u u u + u p(η, γ = ηg in Ω, η + η u u = in Ω, ηc v (η, γγ κ 2 uγ = γp γ (η, γ u u (3.1 + µ 3 (ξ xi ξ u j + ξ xj ξ u i 2 2 i,j=1 + (ν µ( u u 2 + ηk in Ω, u (u, p n = p n on, n u γ = γ on, u = = v in Ω, η = = ρ in Ω, γ = = ϑ in Ω, where u(ξ, = v(x u (ξ,,, γ(ξ, = ϑ(x u (ξ,,, η(ξ, = ρ(x u (ξ,,, g(ξ, = f(x u (ξ,,, k(ξ, = r(x u (ξ,,, u = ξ x ξ {ξ ix ξi }, u (u, p = p1 + D u (u, D u (u = {µ(ξ kxi ξk u j + ξ kxj ξk u i + (ν µδ ij u u} (here he summaion convenion over repeaed indices is assumed and 1 is he uni marix and γ(ξ, = ϑ(x u (ξ,,. Le A = {a ij } be he Jacobi marix of he ransformaion x = X u (ξ,, where a ij =δ ij + ξ j u i (ξ, τ dτ. Assuming ha ξ u,ω M we obain (3.2 < c 1 (1 M 3 de{x ξ } c 2 (1 + M 3,, where c 1, c 2 > are consans and > is sufficienly small. Moreover, ( de A = exp u u dτ = ρ /η.

8 14 E. Zadrzyńska and W. M. Zaj aczkowski Le be deermined (a leas locally by he equaion φ(x, =. hen is described by φ(x(ξ,, = φ(ξ =. hus, we have n(x(ξ,, = xφ(x, x φ(x, and n (ξ = φ(ξ ξ. x=x(ξ, ξ φ(ξ (3.3 Firs we consider he linear problems Lu u µ 2 ξu ν ξ ξ u = F in Ω, D ξ (u n = G on, u = = u in Ω (where D ξ (u = {µ( ξi u j + ξj u i + (ν µδ ij ξk u k } and (3.4 (3.5 and (3.6 We assume γ κ 2 ξγ = K in Ω, n ξ γ = γ on, γ = = γ in Ω. F W 2,1 2 (Ω, G W 3 1/2,3/2 1/4 2 (, D 2 ξ,g 1/4, <, u W 3 2 (Ω K W 2,1 2 (Ω, γ W 3 1/2,3/2 1/4 2 (, D 2 ξ, γ 1/4, <, γ W 3 2 (Ω. Moreover, we assume he following compaibiliy condiions: (3.7 D α ξ (D ξ (u( n G( =, α 1, on, and (3.8 D α ξ (n ξ γ( γ( =, α 1, on. Firs we consider problem (3.3. Define funcions ψ i for i =, 1 by (3.9 ψ i = i u = in Ω. Hence (3.1 ψ = u, ψ 1 = µ u + ν div u + F (. Lemma 3.1. Le Ω R 3 be eiher a halfspace or a bounded domain wih smooh boundary Ω and le. Assume ha (3.11 ψ W 3 2 (Ω, ψ 1 W 1 2 (Ω. hen here exiss v W 4,2 2 (Ω such ha (3.12 i v = = ψ i in Ω (i =, 1

9 Moion of a hea conducing fluid 141 and (3.13 v 4,Ω c( ψ 3,Ω + ψ 1 1,Ω, where c > is a consan independen of. P r o o f. Using he Hesenes Whiney mehod (see [1] we can exend ψ and ψ 1 o funcions ψ W 3 2 (R 3 and ψ 1 W 1 2 (R 3 such ha ψ 3,R 3 c ψ 3,Ω, ψ 1 1,R 3 c ψ 1 1,Ω, where c = c(ω. hen from [16] (Lemma 6.5 we deduce ha here exiss ṽ W 4,2 2 (R 3 R 1 + such ha (3.14 i ṽ = = ψ i, i =, 1, and (3.15 ṽ 4,R 3 R 1 + c( ψ 3,R 3 + ψ 1 1,R 3 c( ψ 3,Ω + ψ 1 1,Ω. herefore v = ṽ Ω saisfies condiions (3.12 and (3.13. Now we prove he following heorem. heorem 3.1. Le W 4 1/2 2 and le assumpions (3.5 and (3.7 be saisfied ( <. hen here exiss a soluion of (3.3 such ha u W 4,2 2 (Ω and (3.16 u 4,Ω c( ( F 2,Ω + G 3 1/2, + D 2 ξ,g 1/4, + u( 3,,Ω, where c is an increasing coninuous funcion of and (3.17 u( 3,,Ω = u 3,Ω + u = 1,Ω. P r o o f. Inroduce he funcion (3.18 w = u v, where v is he funcion from Lemma 3.1. hen insead of (3.3 we obain he problem w µ w ν div w = F Lv f in Ω, (3.19 D ξ (w n = G D ξ (v n g on, w = = in Ω, where in view of he compaibiliy condiion (3.7 we have (3.2 f( = g( =. I is sufficien o consider problem (3.19 in Ω R R 3 + [, ] because using a pariion of uniy and appropriae norms (see [8], ecs. 2, 21 we obain he exisence and he appropriae esimae of soluions of (3.19 in a bounded domain Ω.

10 142 E. Zadrzyńska and W. M. Zaj aczkowski hus, consider problem (3.19 in R and exend funcions f, g, w by zero for < o funcions f 1, g 1, w 1. hen insead of (3.19 we ge he following boundary value problem: (3.21 Lw 1 = f 1 in D 4 ( = R 3 + (, ], D ξ (w 1 n = g 1 in Ẽ3( = R 2 (, ]. Nex using he Hesenes Whiney mehod exend f 1 and g 1 o funcions f 2 and g 2 defined on R 3 + (, and R 2 (,, respecively. hen insead of (3.21 we have he problem (3.22 and he esimae Lw 2 = f 2 in D 4 = R 3 + (,, D ξ (w 2 n = g 2 in Ẽ3 = R 2 (,, (3.23 f 2 2, D 4 c f 1 2, D 4 (. Nex we exend f 2 by he Hesenes Whiney mehod o a funcion f 3 W 4,2 2 (R 4 such ha (3.24 f 3 2,R 4 c f 2 2, D 4. Consider now he sysem (3.25 Lw 3 = f 3 in R 4. By poenial echniques (see [8], ecions 12 and 21 and (3.23, (3.24 here exiss a soluion w 3 W 4,2 2 (R 4 of (3.25 and (3.26 w 3 4,R 4 c( f 2,R, where c( is an increasing funcion of. Inroduce he funcion (3.27 w 4 = w 2 w 3. By (3.22 and (3.25 we have (3.28 Lw 4 = in D 4, D ξ (w 4 n = g 2 D ξ (w 3 n g 3 in Ẽ3. Again by poenial echniques here exiss a soluion w 4 W 4,2 2 ( D 4 of (3.28 and w 4 4, D 4 c[g 3 ] 3 1/2,2,Ẽ3 c([g 2 ] 3 1/2,2,Ẽ3 + w 3 4, D 4 c( ([g 1 ] 3 1/2,2, Ẽ 3 ( + f 2,R, where we used (3.26 and he Hesenes Whiney mehod for g 1. Hence

11 Moion of a hea conducing fluid 143 (3.29 w 4 4,R c( ([g 1 ] 3 1/2,2,Ẽ3 ( + f 2,R. Furher, using Lemma 2.3 we have (3.3 [g 1 ] = 3 1/2,2,Ẽ3 [D α ( ξ g 1 ] + [ 1/2,2,Ẽ3 (,ξ g 1 ] 1/2,2,Ẽ3 (,ξ α =2 + [Dξ α g 1 ] + [ 1/4,2,Ẽ3 (, g 1 ] 1/4,2,Ẽ3 (, α =2 { c [Dξ α g] 1/2,2,R 2 [, ],ξ + [ g] 1/2,2,R 2 [, ],ξ α =2 + [Dξ α g] 1/4,2,R 2 [, ], + [ g] 1/4,2,R 2 [, ], α =2 ( + D 2 ξ, g 2 2,R 2 1/2 1/2 } d c( g 3 1/2, + D 2 ξ,g 1/4,, where = R 2 [, ]. aking ino accoun he righ-hand side of (3.19 and (3.18, (3.21, (3.22, (3.25 (3.3 we conclude ha here exiss a soluion u W 4,2 2 (Ω of (3.3 saisfying (3.31 u 4,R where c( ( F 2,R + v 4,R + G 3 1/2, + D 2 ξ,g 1/4, + v ξ n 3 1/2, + D 2 ξ,(v ξ n 1/4,, By Lemma 3.1 we have v ξ n = 3 i,j,k=1 v iξj n k. (3.32 v 4,R c( φ 3,R φ 1 1,R 3 + and (3.33 v ξ n 3 1/2, ṽ ξ n 3 1/2, (, c ṽ ξ 3,R 3 + (, c v 4,R c( φ 3,R φ 1 1,R 3 +, where we have used he fac ha W 4 1/2 2. I remains o esimae D 2 ξ, (v ξn 1/4,. We have

12 144 E. Zadrzyńska and W. M. Zaj aczkowski (3.34 D 2 ξ,(v ξ n 1/4, = ( D 2 ξ n D 1 ξ v + D1 ξ n D 2 ξ v + n D 3 ξ v + n D 1 ξ v 2 2, 1/2 1/2 d ( c D 2 ξ n 2 4, D1 ξ v 2 4, + D2 ξ v 2 2, + D3 ξ v 2 2, + D 1 ξ v 2 2, 1/2 1/2 d, where we have used he fac ha W 4 1/2 2 and Lemma 2.1, and where he producs are undersood in he sense of (2.2. Nex, by Lemma 2.1 we ge D 2 ξ n 2 4, D1 ξ v 2 4, 1/2 d c Hence in view of Lemma 2.4, (3.34 yields D 1 ξ v 2 1, 1/2 d. Dξ,(v 2 ξ n 1/4, c ( c 1 [ D 1 ξṽ 2 2, + D2 ξṽ 2 2, + D3 ξṽ 2 2, + D1 ξ ṽ 2 2, 1/2 d d ( D 1 ξ ṽ(ξ, D 1 ξṽ(ξ, 2 2, 1+1/2 1/2 d herefore + D2 ξṽ(ξ, D2 ξṽ(ξ, 2 2, + D ξṽ(ξ, 1 D ξṽ(ξ, 1 2 2, 1+1/2 1+1/2 + D3 ξṽ(ξ, D3 ξṽ(ξ, 2 ] 1/2 2,. 1+1/2 (3.35 D 2 ξ,(v ξ n 1/4, c v 4,R, where we have used heorem 5.1 from [8]. aking ino accoun (3.31 (3.33 and (3.35 we ge (3.16. his complees he proof of he heorem. In he same way we can prove heorem 3.2. Le W 4 1/2 2 and le assumpions (3.6 and (3.8 be saisfied ( <. hen here exiss a soluion of (3.4 such ha γ W 4,2 2 (Ω and (3.36 γ 4,Ω c( ( K 2,Ω + γ 3 1/2, + D 2 ξ, γ 1/4, + γ( 3,,Ω,

13 where c is an increasing funcion of and Moion of a hea conducing fluid 145 (3.37 γ( 3,,Ω = γ 3,Ω + 1 γ = 1,Ω. (3.38 and (3.39 Now we have o consider he following problems: ηu µ 2 ξu ν ξ ξ u = F in Ω, D ξ (u n = G on, u = = u in Ω ηc v (η, βγ κ 2 ξγ = K in Ω, n ξ γ = γ on, γ = = γ in Ω. Firs we consider (3.38. he following heorem is proved in [15]. heorem 3.3. Assume ha W 4 1/2 2, F W 2,1 2 (Ω, G W 3 1/2,3/2 1/4 2 (, Dξ, 2 G 1/4, <, u W2 3 (Ω, η L (, ; Γ 2,1 (Ω C α (Ω (α (, 1, 1/η L (Ω. Moreover, le he compaibiliy condiion (3.7 be saisfied. hen here exiss a unique soluion u W 4,2 2 (Ω o problem (3.38 saisfying he esimae (3.4 u 4,Ω φ 1 ( 1/η,Ω, η,ω, [ F 2,Ω + G 3 1/2, + D 2 ξ,g 1/4, + φ 2 ( η 2,,,Ω, η Cα (Ω u 2,Ω ] + φ 3 ( 1/η,Ω, u( 3,,Ω, where φ i (i = 1, 2, 3 are nonnegaive increasing coninuous funcions of heir argumens. Now we consider problem (3.39. Assume ha η L (, ; Γ 2,1 (Ω and β W 4,2 2 (Ω. Applying heorems 1.2 and 1.4 of [3] we find ha η CB (Ω and β CB (Ω. herefore, here exiss a bounded domain V R 2 such ha (η(ξ,, β(ξ, V for any (ξ, Ω. Lemma 3.2. Assume ha η L (, ; Γ 2,1 (Ω C α (Ω (where α (, 1/2, 1/η L (Ω, η >, β W 4,2 2 (Ω, 1/β L (Ω, β >, c v C 2 (R 2 +, c v >. hen ηc v (η, β L (, ; Γ 2,1 (Ω, ηc v C α (Ω, 1/(ηc v (η, β L (Ω and (3.41 ηc v (η, β 2,,,Ω ψ 1 ( c v C2 (V, 1/c v C (V, η 2,,,Ω, β 4,Ω, β( 3,,Ω, (3.42 ηc v (η, β,ω c v C (V η,ω,

14 146 E. Zadrzyńska and W. M. Zaj aczkowski (3.43 1/(ηc v (η, β,ω 1/c v C (V 1/η,Ω, (3.44 ηc v (η, β C α (Ω ψ 2 ( c v C2 (V, η C α (Ω, β 4,Ω, β( 3,,Ω,, where ψ i (i = 1, 2 are nonnegaive increasing coninuous funcions of heir argumens. P r o o f. By he assumpions we can choose V such ha V R 2 +. hus (3.42 and (3.43 are obviously saisfied. In order o obain (3.41 we calculae he derivaives D α ξ i (ηc v (where α + 2i 2 and nex we apply Lemmas 2.1 and 2.2. o prove (3.44 we also use Lemmas 2.1 and 2.2 and he obolev imbedding heorem. heorem 3.4. Assume ha W 4 1/2 2, K W 2,1 2 (Ω, γ W 3 1/2,3/2 1/4 2 (, γ W2 3 (Ω, η L (, ; Γ 2,1 (Ω, 1/η L (Ω, η C α (Ω ( < α < 1/2, η >, β W 4,2 2 (Ω, 1/β L (Ω, β >, c v C 2 (R 2 +, c v >. Moreover, le he compaibiliy condiion (3.8 be saisfied. hen here exiss a unique soluion γ W 4,2 2 (Ω o problem (3.39 saisfying he esimae γ 4,Ω φ 4 ( c v C (V, 1/c v C (V, η,ω, 1/η,Ω, (3.45 [ K 2,Ω + γ 3 1/2, + D 2 ξ, γ 1/4, + φ 5 ( c v C 2 (V, 1/c v C (V, η 2,,,Ω, η C α (Ω, β 4,Ω, β( 3,,Ω, γ 2,Ω ] + φ 6 ( 1/c v C (V, 1/η,Ω, γ( 3,,Ω, where φ i (i = 4, 5, 6 are nonnegaive increasing coninuous funcions of heir argumens. P r o o f. ince by Lemma 3.2, ηc v (η, β L (, ; Γ 2,1 (Ω, ηc v (η, β C α (Ω (α (, 1/2 and 1/(ηc v (η, β L (Ω, using he same argumen as in heorem 3.3 and inequaliies (3.41 (3.44 we prove he exisence of a unique soluion γ W 4,2 2 (Ω of problem (3.39 saisfying he esimae (3.45. Now consider he problems ηu µ 2 wu ν w w u = F in Ω, (3.46 D w (u n = G on, u = = u in Ω,

15 and (3.47 Moion of a hea conducing fluid 147 ηc v (η, βγ κ 2 wγ = K in Ω, n w γ = γ on, γ = = γ in Ω, where n = n(x w (ξ,,. he following heorem concerns problem (3.46. heorem 3.5. Assume ha W 4 1/2 2, F W 2,1 2 (Ω, G W 3 1/2,3/2 1/4 2 (, u W2 3 (Ω, η L (, ; Γ 2,1 (Ω, 1/η L (Ω, η C α (Ω (α (, 1, w W 4,2 2 (Ω C (, ; Γ 3,3/2 (Ω, <. Moreover, le he compaibiliy condiion (3.7 be saisfied. Assume ha is so small ha (3.48 a ( w 4,Ω + w( 3,,Ω ( φ 7 w ξ,ω d, 1/η,Ω, η,ω, w 4,Ω, w( 3,,Ω, δ, where φ 7 is an nonnegaive nondecreasing coninuous funcion of is argumens, a > is a consan and δ > is sufficienly small. hen here exiss a unique soluion u W 4,2 2 (Ω C (, ; Γ 3,3/2 (Ω o problem (3.46 saisfying he esimae (3.49 u 4,Ω φ 1 ( 1/η,Ω, η,ω, [ F 2,Ω + G 3 1/2, + D 2 ξ,g 1/4, + φ 2 ( η 2,,,Ω, η C α (Ω u 2,Ω ] + φ 3( 1/η,Ω, u( 3,,Ω, where φ i (i = 1, 2 are he funcions from heorem 3.3, and φ 3 is a nonnegaive increasing funcion. P r o o f. In order o prove he exisence of soluions o (3.46 we rewrie i as ηu µ 2 ξu ν ξ ξ u = F + µ( 2 wu 2 ξu + ν( w w u ξ ξ u (3.5 F + F in Ω, D ξ (u n = G + (D ξ (u n D w (u n = D ξ (u (n n + (D ξ (u D w (u n G + G on, u = = u in Ω.

16 148 E. Zadrzyńska and W. M. Zaj aczkowski Using heorem 3.3 we have he following esimae for soluions of (3.5: (3.51 u 4,Ω φ 1 ( 1/η,Ω, η,ω, [ F + F 2,Ω + G + G 3 1/2, + Dξ,G 2 + Dξ, 2 G 1/4, + φ 2 ( η 2,,,Ω, η Cα (Ω u 2,Ω ] + φ 3 ( 1/η,Ω, u( 3,,Ω. Firs we esimae F 2,Ω. By he form of he operaor w we obain (3.52 F 2,Ω c (ξ x ξ 2 u 2 ξu 2,Ω + c ξ x ξ (ξ x ξ u ξ ξ u 2,Ω c (ξ x ξ 2 u 2 ξu 2,Ω + c ξ x ξ (ξ x ξ u 2,Ω + c (ξ 2 x 1D 2 ξu 2,Ω [ f1 c Dξw 2 dτ Dξu 1 + f2 D 1 2,Ω ξw dτ Dξu 2 2,Ω where f i = f i (1 + w ξ dτ (i = 1, 2 are smooh funcions of heir argumens; ξ x = x 1 ξ = x ξ /de{x ξ }, x ξ is he marix of algebraic complemens of {x ξ } and x ξ = 1+ w ξ dτ. he producs on he righ-hand side of (3.52 (and below are in he sense of (2.2. Now, applying Hölder and Minkowski inequaliies and Lemmas we ge (3.53 f 1 Dξw 2 2,Ω dτ u ξ + f2 w ξ dτ Dξu 2 2,Ω ], c[ w 2 4,Ω u 4,Ω + 3/2 w 3 4,Ω u 4,Ω + 1/2 w 4,Ω u 4,Ω where c = c( w ξ,ω d. Nex consider G ( /2, c f 3 + 1/2 w 2 4,Ω ( u 4,Ω + u( 3,Ω + 1/2 ( w 4,Ω + w( 3,Ω ( u 4,Ω + u( 3,Ω ], ( c α +2i 2 w ξ dτ u ξ φξ 3 1/2, Dξ α (f i 3 w ξ dτ u ξ φξ 2,

17 Moion of a hea conducing fluid α +2i=2 + α +2i=2 [ Dξ α (f i 3 [ Dξ α (f i 3 w ξ dτ u ξ φξ ] w ξ dτ u ξ φξ ] 1/2,,ξ 1/4,, where f 3 = f 3 (1 + w ξ dτ is a smooh funcion of is argumens. By he Hölder and Minkowski inequaliies, Lemma 2.1 (inequaliy (2.5 and Lemma 2.2 we have Dξ α (f i 2, ( w ξ dτ u ξ φξ α +2i 2 c[ 1/2 w 4,Ω u 4,Ω + w 2 4,Ω u 4,Ω + 3/2 w 3 4,Ω u 4,Ω + 1/2 w 2 4,Ω ( u 4,Ω + u( 3,,Ω + 1/2 ( w 4,Ω + w( 3,,Ω ( u 4,Ω + u( 3,,Ω ] I, where c = c( w ξ,ω d. Now consider I 1 [ Dξ α (f i 3 α +2i=2 w ξ dτ u ξ φξ ] 1/2,,ξ We shall only esimae some erms of I 1. For example we have I 2 { { c { + c [f 3 ( D2 ξ w dτ2 w ξ dτ u ξ φξ ](ξ [f 3 ( D2 ξ w dτ2 w ξ dτ u ξ φξ ](ξ 2 ξ ξ 3 dξ dξ d [f 3 ( w ξ dτ](ξ [f 3 ξ ξ 3 Dξw 2 dτ 4 Dξw 2 dτ 2 w ξ dτ](ξ 2 w ξ dτ 2 u ξ 2 φ ξ 2 dξ dξ d [D2 ξ w(ξ D2 ξ w(ξ ] dτ 2 ξ ξ 3 w ξ dτ 2 u ξ 2 φ ξ 2 dξ dξ d } 1/2 } 1/2., } 1/2

18 15 E. Zadrzyńska and W. M. Zaj aczkowski { + c { + c { + c [w ξ(ξ w ξ (ξ ] dτ 2 ξ ξ 3 u ξ (ξ u ξ (ξ 2 ξ ξ 3 φ ξ (ξ φ ξ (ξ 2 ξ ξ 3 Using he aylor formula we ge (3.56 I 2 c { + c { + c { + c { + c { Dξw 2 dτ 4 Dξw 2 dτ 2 D 2 ξw dτ 4 Dξw 2 dτ 4 D2 ξ w( ξ dτ 2 ξ ξ Dξw 2 dτ 4 u ξ 2 φ ξ 2 dξ dξ d w ξ dτ 2 u ξ 2 φ ξ 2 dξ dξ d [D2 ξ w(ξ D2 ξ w(ξ ] dτ 2 ξ ξ 3 w ξ dτ 2 u ξ 2 φ ξ 2 dξ dξ d D2 ξ w( ξ dτ 2 ξ ξ Dξ 2u( ξ 2 ξ ξ Dξ 2 φ( ξ 2 ξ ξ I I I I I 5 2, where ξ = θξ + (1 θξ, < θ < 1. Obviously, we have For I 2 2, we obain I 2 2 c { sup D 2 ξw dτ 4 Dξw 2 dτ 4 w ξ dτ 2 φ ξ 2 dξ dξ d } 1/2 } 1/2 w ξ dτ 2 1/2 u ξ 2 dξ dξ d}. } 1/2 } 1/2 Dξw 2 dτ 4 u ξ 2 φ ξ 2 dξ dξ d I 1 2 c 2 w 4 4,Ω u 4,Ω. Dξw 2 dτ 2 sup w ξ dτ } 1/2 } 1/2 w ξ dτ 2 φ ξ 2 dξ dξ d w ξ dτ 2 u ξ 2 dξ dξ d 2 sup u ξ 2 sup φ ξ 2 } 1/2

19 { Moion of a hea conducing fluid 151 [D2 ξ w(ξ D2 ξ w(ξ ] dτ 2 } 1/2 ξ ξ 3 dξ dξ d c 1/2 w 4,Ω 1/2 w 4,Ω 1/2 Dξ 2 w(ξ D2 ξ w(ξ 2 } 1/2 ξ ξ 3 dξ dξ d u 4,Ω c 3/2 w 3 4,Ω u 4,Ω, where we have used he fac ha Dξ 2w W 1/2,1/4 2 ( and Lemma 2.1 (inequaliy (2.5. Nex we esimae I 3 2 c { sup sup Dξw 2 dτ D 2 ξw( ξ dτ c 3/2 w 3 4,Ω u 4,Ω. 4 sup 2( u ξ 2 sup φ ξ 2 dξdξ ξ ξ } 1/2 d he oher erms boh of (3.56 and of (3.55 can be esimaed in he same way. herefore, by he above consideraions α +2i=2 α +2i=2 [ Dξ α (f i 3 w ξ dτ u ξ φξ ] where I is defined in (3.55. Consider now [ ] Dξ α (f i 3 w ξ dτ u ξ φξ where A(ξ,, [ c Dξ, (f 2 3 D 2 ξ, ( f 3 1/2,,ξ 1/4,, (1 + w 4,Ω I, A(ξ,, ] 1/2 dξ d d I 3/2 3, w ξ dτ u ξ φξ ( w ξ dτ u ξ φξ ( 2.

20 152 E. Zadrzyńska and W. M. Zaj aczkowski As before we only esimae some erms of I 3. e ( B(ξ,, We have [ (3.57 I 5 ( c ( + c ( + c ( + c f 3 ( ( f 3 ( 2 Dξw 2 dτ D 2 ξw dτ 2 B(ξ,, ] 1/2 dξ d d 3/2 f 3 ( w ξ dτ f 3 ( w ξ dτ 2 3/2 Dξw 2 dτ 4 Dξw 2 dτ 2 I I I I 4 4. w ξ dτu ξ φξ ( w ξ dτ 2 1/2 u ξ 2 φ ξ 2 dξ d d D2 ξ w dτ D2 ξ w dτ 2 3/2 w ξ dτ 2 1/2 u ξ 2 φ ξ 2 dξ d d u ξ ( u ξ ( 2 3/2 Dξw 2 dτ 4 w ξ dτ w ξ dτ 2 3/2 Using he aylor formula we obain ( I4 2 c D2 ξ w dτ D2 ξ w dτ 2 3/2 ( c Dξw 2 dτ 2 1/2( w ξ dτu ξ φξ ( 2. w ξ dτ 2 1/2 u ξ 2 φ ξ 2 dξ d d Dξw( 2 2 dξ w ξ dτ 2 1/2 φ ξ 2 dξ d d Dξw 2 dτ 4 1/2 u ξ 2 φ ξ 2 dξ d d

21 sup Moion of a hea conducing fluid 153 Dξw 2 dτ 2 sup ( c w 2 4,Ω sup w( 2 3,Ω w ξ dτ 2 sup u ξ 2 sup φ ξ 2 d d 1/2 1/2 sup u ξ 2 d d 1/2 c 5/4 w 2 4,Ω ( w 4,Ω + w( 3,,Ω u 4,Ω, where = θ + (1 θ, < θ < 1. In he same way he oher erms boh of (3.57 and of α +2i=2 [ Dξ α (f i 3 w ξ dτ u ξ φξ ] can be esimaed. hus, from he above consideraions we obain (3.58 1/4,, G 3 1/2, 1/2 ( w 4,Ω + w( 3,,Ω ( φ w,ω d, w 4,Ω, w( 3,,Ω, ( u 4,Ω + u( 3,,Ω, where φ is a nonnegaive nondecreasing coninuous funcion of is argumens. I remains o esimae D 2 ξ, G 1/4,. We ge (3.59 D 2 ξ, G 1/4, = ( D 2 ξ, (f 3 w ξ dτ u ξ φξ 2 2, 1/2 1/2 d c[ 1/4 w 4,Ω u 4,Ω + 3/4 w 4,Ω u 4,Ω + 5/4 w 3 4,Ω u 4,Ω + 1/4 w 4,Ω ( u 4,Ω + u( 3,,Ω + 1/4 ( w 4,Ω + w( 3,,Ω ( u 4,Ω + u( 3,,Ω ]. By (3.51 (3.59 we ge u 4,Ω φ 1 ( 1/η,Ω, η,ω, [ F 2,Ω + G 3 1/2, + D 2 ξ,g 1/4, + φ 2 ( η 2,,,Ω, η Cα (Ω u 2,Ω ] + φ 3 ( 1/η,Ω, u( 3,,Ω + a ( w 4,Ω + w( 3,,Ω φ 7 ( 1/η,Ω, η,ω, w,ω d, w 4,Ω, w( 3,,Ω, ( u 4,Ω + u( 3,,Ω, where a > is a consan.

22 154 E. Zadrzyńska and W. M. Zaj aczkowski herefore if δ > is sufficienly small (3.49 follows from (3.48. o prove he exisence and uniqueness of soluions o problem (3.46 we use he mehod of successive approximaions. Pu u m in he righ-hand side of (3.5 in place of u and replace u by u m+1 in he lef-hand side of (3.5. hen by he conracion heorem we have he exisence of soluions of problem (3.46 for sufficienly small. herefore he heorem is proved. In he same way he following heorem can be proved: heorem 3.6. Assume ha W 4 1/2 2, K W 4,2 2 (Ω, γ W 3 1/2,3/2 1/4 2 (, D ξ, γ 2 1/4, <, γ W2 3 (Ω, η L (, ; Γ 2,1 (Ω, 1/η L (Ω, η >, η C α (Ω (α (, 1/2, β W 4,2 2 (Ω, 1/β L (Ω, β >, c v C 2 (R 2 +, c v >, w W 4,2 2 (Ω C (, ; Γ 3,3/2 (Ω, <. Moreover, le he compaibiliy condiion (3.8 be saisfied. Assume ha is so small ha (3.6 a ( w 4,Ω + w( 3,,Ω φ 8 ( w ξ,ω d, c v C2 (V, 1/c v C (V, 1/η,Ω, η,ω, β 4,Ω, β( 3,,Ω, w 4,Ω, w( 3,,Ω, δ, where φ 8 is a nonnegaive nondecreasing coninuous funcion of is argumens, a > is a consan and δ > is sufficienly small. hen here exiss a unique soluion γ W 4,2 2 (Ω C (, ; Γ 3,3/2 (Ω o problem (3.47 saisfying he esimae γ 4,Ω φ 4 ( c v C 2 (V, 1/c v C (V, 1/η,Ω, η,ω, (3.61 [ K 2,Ω + γ 3 1/2, + D 2 ξ, γ 1/4, + φ 5 ( c v C2 (V, 1/c v C (V, η 2,,,Ω, η C α (Ω, β 4,Ω, β( 3,,Ω γ 2,Ω ] + φ 6( 1/c v C (V, 1/η,Ω, γ( 3,,Ω, where φ i (i = 4, 5 are he funcions from heorem 3.4, and φ 6 is a nonnegaive increasing funcion. Now consider he coninuiy equaion ( Inegraing i we have [ ] (3.62 η(ξ, = ϱ (ξ exp u u(ξ, τ dτ. he following lemma holds (see [15].

23 Moion of a hea conducing fluid 155 Lemma 3.3. Le ϱ W 3 2 (Ω, u C (, ; W 3 2 (Ω W 4,2 2 (Ω. hen he soluion (3.62 of he coninuiy equaion saisfies η C (, ; W 3 2 (Ω, η C (, ; W 2 2 (Ω L 2 (, ; W 3 2 (Ω, η L 2 (, ; W 1 2 (Ω and he following esimaes hold: (3.63 sup η 3,Ω ψ 1 ( 1/2 u 4,Ω ( 1/2 u 4,Ω + 1 ϱ 3,Ω, (3.64 η L2 (, ;W 3 2 (Ω ψ 2 ( 1/2 u 4,Ω 1/2 u 4,Ω ϱ 3,Ω, (3.65 η L2 (, ;W 1 2 (Ω ψ 3 ( 1/2 u 4,Ω ( 1/2 + 1 u 4,Ω ϱ 3,Ω, (3.66 sup (3.67 η 2,Ω ψ 4 ( 1/2 u 4,Ω sup u 3,Ω ϱ 3,Ω, 1/η,Ω ψ 5 ( 1/2 u 4,Ω 1/ϱ,Ω, η,ω ψ 6 ( 1/2 u 4,Ω ϱ,ω, (3.68 η Cα (Ω ψ 7 ( 1/2 u 4,Ω [ ψ 8 ( 1/2 u 4,Ω + 1 α ( u 4,Ω + u( 3,Ω ] ϱ Cα (Ω, where ψ i (i = 1,..., 8 are nonnegaive nondecreasing coninuous funcions. Now we are able o prove he exisence of soluion o problem (3.1. heorem 3.7. Le W 4 1/2 2, f C 2,1 (R 3 [, ], r C 2,1 (R 3 [, ], ϑ C 2,1 (R 3 [, ], v W 3 2 (Ω, ϑ W 3 2 (Ω, 1/ϑ L (Ω, ϑ >, ϱ W 3 2 (Ω, 1/ϱ L (Ω, ϱ >, c v C 2 (R 2 +, c v >, p C 3 (R 2 +. Moreover, assume ha he following compaibiliy condiions are saisfied: (3.69 D α ξ (D ξ (v n p(ϱ, ϑ n + p n =, α 1, on, and (3.7 D α ξ (n ξ ϑ ϑ(ξ, =, α 1, on. Le > be so small ha < c 1 (1 CK 3 de{x ξ } c 2 (1 + CK 3 (where x(ξ, = ξ + u (ξ, dτ for, u is given by (3.74, K is given by (3.91 and he consan C = C(K, which is a nondecreasing coninuous funcion of K, is given by (3.94. hen here exiss wih < such ha for here exiss a unique soluion (u, γ, η W 4,2 2 (Ω W 4,2 2 (Ω C (, ; Γ 3,3/2 (Ω of problem (3.1 and sup u 4,Ω + γ 4,Ω CK, η 3,Ω + sup η 2,Ω + η L2 (, ;W 3 2 (Ω + η L2 (, ;W 1 2 (Ω Φ 1 (, a K ϱ 3,Ω,

24 156 E. Zadrzyńska and W. M. Zaj aczkowski 1/η,Ω + η,ω Φ 2 ( 1/2 K 1/ϱ,Ω + Φ 3 ( 1/2 K ϱ,ω, where Φ i (i = 1, 2, 3 are increasing coninuous funcions of heir argumens, a >. P r o o f. We prove he exisence of soluions of problem (3.1 applying he mehod of successive approximaions. o do his consider he problems η m u m+1 µ 2 u m u m+1 ν um um u m+1 (3.71 = um p(η m, γ m + η m g in Ω, D um (u m+1 n(u m = [p(η m, γ m p ]n(u m on, u m+1 = = v in Ω ; η m c v (η m, γ m γ m+1 κ 2 u m γ m+1 = γ m p γ (η m, γ m um u m ( µ 3 (ξ xi ξ u mj + ξ xj ξ u mi 2 2 i,j=1 + (ν µ( um u m 2 + η m k in Ω, n(u m um γ m+1 = γ on, γ m+1 = = ϑ (where γ(ξ, = ϑ(x um (ξ,, = ϑ(ξ + u m dτ, and (3.73 η m + η m um u m = in Ω, η m = = ϱ in Ω. For u we ake a funcion from Lemma 3.1 such ha (3.74 i u = = ψ i in Ω (i =, 1 and (3.75 u 4,Ω c( ψ 3,Ω + ψ 1 1,Ω, where ψ i = i u = are calculaed from problem (3.1, i.e. in Ω (3.76 ψ = v, ϱ ψ 1 µ 2 ξv ν ξ ξ v + ξ p(ϱ, ϑ = ϱ g(. imilarly, for γ we ake a funcion from Lemma 3.1 such ha (3.77 i γ = = σ i in Ω (i =, 1 and (3.78 γ 4,Ω c( σ 3,Ω + σ 1 1,Ω,

25 where σ i = i γ = (i =, 1, i.e. σ = ϑ, Moion of a hea conducing fluid 157 (3.79 η c v (η, ϑ σ 1 κ 2 ξϑ = ϑ p γ (η, ϑ ξ v + µ 3 ( ξi v j + ξj v i 2 + (ν µ( ξ v 2 + η k(, 2 i,j=1 and η is a soluion of he problem (3.8 η + η u u =, η = = ϱ. Assume ha u m W 4,2 2 (Ω and γ m W 4,2 2 (Ω. hen by Lemma 3.3, 1/η m L (Ω, η m > and η m C α (Ω. Nex, since γ m ( = τ γ m (τ dτ + γ m ( we have γ m ( ϑ 1/2( τ γ m 2 dτ 1/2. hus, from he assumpion ha 1/ϑ L (Ω and ϑ > i follows ha for sufficienly small, 1/γ m L (Ω and γ m >. Now assume ha inequaliies (3.48 and (3.6 wih w = u m, η = η m and β = γ m are saisfied wih sufficienly small δ. hen by heorem 3.5 here exiss a unique soluion u m+1 W 4,2 2 (Ω (where = (δ is also small of (3.71 such ha (3.81 u m+1 4,Ω φ 1 ( 1/η m,ω, η m,ω, [ um p(η m, γ m 2,Ω + η m g 2,Ω + [p(η m, γ m p ]n(u m 3 1/2, + D 2 ξ,{[p(η m, γ m p ]n(u m } 1/4, + φ 2 ( η m 2,,,Ω, η m C α (Ω u m 2,Ω ] + φ 3( 1/η m,ω, u m ( 3,,Ω. imilarly, by heorem 3.6 here is a unique soluion γ m+1 W 4,2 2 (Ω ( = (δ of (3.72 such ha (3.82 γ m+1 4,Ω φ 4 ( c v C 2 (V m, 1/c v C (V m, 1/η m,ω, η m,ω, [ γ m p γ (η m, γ m um u m 2,Ω + ( um u m 2 2,Ω + η m k 2,Ω + γ 3 1/2, + D 2 ξ, γ 1/4,

26 158 E. Zadrzyńska and W. M. Zaj aczkowski + φ 5 ( c v C 2 (V m, 1/c v C (V m, η m 2,,,Ω, η m Cα (Ω, γ m 4,Ω, γ m ( 3,,Ω γ m 2,Ω ] + φ 6( 1/c v C (V m, 1/η m,ω, γ m ( 3,,Ω, where V m R 2 + is a bounded domain such ha (η m (ξ,, γ m (ξ, V m for any (ξ, Ω. In order o esimae he erms on he righ-hand sides of (3.81 and (3.82 we use he same mehods as in heorem 3.5. hus, we ge (3.83 um p(η m, γ m 2,Ω = ξ x ξ p(η m, γ m 2,Ω ξ x (p η Dξη 1 m + p γ Dξγ 1 m 2,Ω Dξ α [ i f 1 (p η Dξη 1 m + p γ Dξγ 1 m ] 2,Ω α +2i 2 c f 1 C 2 (G m p C 2 (V m α 1( a (sup a ( u m 4,Ω + sup η m 1,Ω + sup η m 3,Ω, u m ( 3,,Ω + γ m 4,Ω + sup γ m ( 3,,Ω, a 1, (3.84 γ m p γ (η m, γ m um u m 2,Ω = γ m p γ (η m, γ m f 1 D 1 ξu m 2,Ω c f 1 C2 (G m p C 3 (V m α 2( a (sup a ( u m 4,Ω + sup and (3.85 ( um u m 2 2,Ω + η m 1,Ω + sup η m 3,Ω, u m ( 3,,Ω + γ m 4,Ω + sup γ m ( 3,,Ω, a 2 3 (ξ xi ξ u mj + ξ xj ξ u mi 2 2,Ω i,j=1 ( f 2 ξ u m 2 2,Ω c f 2 C2 (G m α 3( a ( u m 4,Ω + sup u m ( 3,,Ω, a 3. In (3.83 (3.85, fi = f i ( D1 ξ u m dτ (i = 1, 2 are smooh funcions, G m R 9 is a bounded domain such ha D1 ξ u m dτ G m for any (ξ, Ω, α i (i = 1, 2, 3 are nonnegaive increasing coninuous funcions of heir argumens which are polynomials such ha α i (,, =, a > and a i > (i = 1, 2, 3 are consans. Nex, we obain (3.86 η m g 2,Ω c f C 2,1 (Q m α 4( a (sup a ( u m 4,Ω η m 3,Ω + sup η m 1,Ω, u m ( 3,,Ω, a 4, + sup

27 Moion of a hea conducing fluid 159 (3.87 η m k 2,Ω c r C 2,1 (Q m α 5( a (sup and (3.88 γ 3 1/2, + D 2 ξ, γ 1/4, a ( u m 4,Ω c ϑ C 2,1 (Q m α 6( a ( u m 4,Ω η m 3,Ω + sup η m 1,Ω, u m ( 3,,Ω, a 5 + sup + sup u m ( 3,,Ω, a 6, where Q m R 3 [, ] is a bounded domain such ha (ξ+ u m dτ, Q m for any (ξ, Ω, α i (i = 4, 5, 6 are nonnegaive increasing coninuous funcions of heir argumens which are polynomials such ha α i (,, =, and a i > (i = 4, 5, 6 are consans. Finally, we have (3.89 [p(η m, γ m p ]n(u m 3 1/2, + Dξ,{[p(η 2 m, γ m p ]n(u m } 1/4, { c Dξ α {[p(η i m, γ m p ] f} 2, α +2i 2 + α +2i=2 + α +2i=2 ( + [D α ξ i {[p(η m, γ m p ] f}] 1/2,,ξ [D α ξ i {[p(η m, γ m p ] f}] 1/4,, D 2 ξ, {[p(η m, γ m p ] f} 2 2, 1/2 c f C 3 (G m p C 3 (G m α 7( a (sup a ( u 4,Ω + sup 1/2 } d η m 3,Ω + sup η m 1,Ω, u m ( 3,,Ω + γ m 4,Ω + sup γ m ( 3,,Ω, a 7, where α 7 has he same properies as α 1 and α 2, a 7 > is a consan, and f = f( D1 ξ u m dτ is a smooh funcion. Le y m ( = u m 2 4,Ω + γ m 2 4,Ω + sup u m ( 2 3,,Ω + sup γ m ( 2 3,,Ω, τ (, τ (, b = ϱ 3,Ω + 1/ϱ,Ω + ϱ,ω + u m ( 3,,Ω + γ m ( 3,,Ω, c m = p C 3 (V m + f C 2,1 (Q m + r C 2,1 (Q m + ϑ C 2,1 (Q m + c v C 2 (V m + 1/c v C (V m + f 2 C2 (G m + f i C2 (G m. i=1

28 16 E. Zadrzyńska and W. M. Zaj aczkowski Hence, aking ino accoun inequaliies (3.81 (3.89 and using Lemma 3.3 we obain y m+1 ( β 1 ( a y m (, b, c m, + β 2 ( a y m (, y m (, b, c m y m+1 (τ dτ, where β i (i = 1, 2 are nonnegaive nondecreasing coninuous funcions of heir argumens and β 1 (, b, c m, = [φ 3( ψ 5 ( 1/ϱ,Ω, + φ 6(c m, ψ 5 ( 1/ϱ,Ω, ]. In view of he Gronwall lemma we have (3.9 y m+1 ( β 1 ( a y m (, b, c m, Using (3.75 and (3.78 we ge exp[β 2 ( a y m (, y m (, b, c m ]. (3.91 y ( c( ψ 2 3,Ω + ψ 1 2 1,Ω + σ 2 3,Ω + σ 1 2 1,Ω < cb K. Assume ha (3.92 y m ( CK (where C > is a consan which will be chosen laer, and >. We shall prove ha (3.93 c m c, where c is a consan independen of m and C. In fac, by Lemma 3.3 and (3.9 we have η m,ω ψ 6 ( 1/2 CK ϱ,ω K for sufficienly small = (CK (where K is a consan independen of CK. imilarly γ m,ω ϑ + 1/2 CK K for sufficienly small. Hence here exiss a bounded domain V R 2 + such ha V m V for any m. hus, we proved ha p C 3 (V m + c v C 2 (V m is esimaed by a consan independen of m and K. In he same way we can prove ha here exis bounded domains G R 9 and Q R 3 (, such ha G m G and Q m Q for any m. hus we have proved (3.93. Now inequaliies (3.92 (3.93 yield ha for we have y m+1 ( β 1 ( a CK, b, c, exp[β 2 ( a CK, CK, b, c] CK, if C = C(K is chosen so large ha (3.94 β 1 (, b, c, < CK

29 Moion of a hea conducing fluid 161 and if is sufficienly small (C(K in (3.94 is a nondecreasing coninuous funcion of K. In his way we have shown ha here exiss a sufficienly large C = C(K (saisfying (3.94 such ha C(K is a nondecreasing coninuous funcion of K, and a sufficienly small > such ha for and for m =, 1,... we have (3.95 y m ( CK. (By (3.95 and (3.93 inequaliies (3.48 and (3.6 wih w = u m, η = η m and β = γ m are saisfied wih sufficienly small δ and sufficienly small independen of m. Now we prove he convergence of he sequence {u m, γ m, η m }. Consider he following sysem of problems for he differences U m+1 = u m+1 u m, Γ m+1 = γ m+1 γ m, H m = η m η m 1 : η m U m+1 µ 2 u m U m+1 ν um um U m+1 = H m u m µ( 2 u m 2 u m 1 u m ν( um um um 1 um 1 u m + um p(η m, γ m um 1 p(η m 1, γ m 1 + H m g (3.96 F 1 + F 2, D um (U m+1 n(u m = [D um (u m n(u m D um 1 (u m n(u m 1 ] + [p(η m, γ m n(u m p(η m 1, γ m 1 n(u m 1 ] p [n(u m n(u m 1 ] G 1 + G 2, U m+1 = = ; η m c v (η m, γ m Γ m+1 κ 2 u m Γ m+1 = H m c v (η m, γ m γ m + η m 1 γ m [c v (η m 1, γ m 1 c v (η m, γ m ] ( µ 3 [(ξ xi ξ u mj + ξ xj ξ u mi 2 2 i,j=1 (ξ xi ξ u m 1,j + ξ xj ξ u m 1,i 2 ] + (ν µ[( um u m 2 ( um 1 u m 1 2 ] γ m p γ (η m, γ m um u m + γ m 1 p γ (η m 1, γ m 1 um 1 u m 1 + H m k I 1 + I 2,

30 162 E. Zadrzyńska and W. M. Zaj aczkowski (3.97 [con.] n(u m um (Γ m+1 = [n(u m um γ m n(u m 1 um 1 γ m ] J, Γ m = = ; (3.98 H m + H m div um u m = η m 1 (div um u m div um 1 u m 1, H m = =, where div ui u i = ui u i, F 2 = H m u m + H m g + um p(η m, γ m um 1 p(η m 1, γ m 1, G 2 = p(η m, γ m n(u m p(η m 1, γ m 1 n(u m 1, I 2 = H m c v (η m, γ m γ m + η m 1 γ m [c v (η m 1, γ m 1 c v (η m, γ m ] + H m k + γ m 1 p γ (η m 1, γ m 1 um 1 u m 1 γ m p γ (η m, γ m um u m and F 1, G 1, I 1 are deermined by he remaining erms on he righ-hand sides. Applying heorems 3.5, 3.6 and Lemma 3.3 o problems (3.96 and (3.97 respecively and using inequaliies (3.93, (3.95 we obain (3.99 U m+1 4,Ω + sup U m+1 3,,Ω + Γ m+1 4,Ω + sup Γ m+1 3,,Ω φ(, K [ U m+1 2,Ω + F 1 2,Ω + F 2 2,Ω + I 1 2,Ω + I 2 2,Ω + G 1 3 1/2, + G 2 3 1/2, + D 2 ξ,g 1 1/4, + D 2 ξ,g 2 1/4, + J 3 1/2, + D 2 ξ,j 1/4, ], where φ is an nonnegaive increasing coninuous funcion of is argumens. o esimae he righ-hand sides of (3.99 we shall consider he following funcions conneced wih he qualiaive forms of F i, I i, G i (i = 1, 2 and J : F 1 = f 1 (1 + Dξũ 1 m dτ DξU 1 m dτ Dξu 2 m (3.1 + f 2 (1 + D 1 ξu m dτ, 1 + Dξũ 1 m dτ D 1 ξu m dτ D 2 ξũ m dτ D 1 ξu m

31 + f 3 (1 + F 2 = H m Moion of a hea conducing fluid 163 D 1 ξũ m dτ, 1 + D 2 ξu m 1 dτ D 1 ξu m ; 3 u mi + H m i=1 + f 4 (1 + + f 5 (1 + 3 i=1 Dξũ 1 m dτ g i Dξu 1 m 1 dτ D 1 ξu m dτ D 1 ξu m dτ [p η (η m, γ m D 1 ξη m + p γ (η m, γ m D 1 ξγ m ] Dξu 1 m 1 dτ [p ηη ( η m, γ m Dξ 1 η m H m + p ηγ ( η m, γ m D 1 ξγ m H m + p γη ( η m, γ m D 1 ξ η m Γ m (3.1 [con.] + p γγ ( η m, γ m D 1 ξ γ m Γ m + p η ( η m, γ m D 1 ξh m + p γ ( η m, γ m D 1 ξγ m ] ; I 1 = f 6 (1 + + f 7 (1 + D 1 ξu m + f 8 (1 + + f 9 (1 + Dξu 1 m dτ Dξu 1 m DξU 1 m D 1 ξu m dτ, 1 + D 1 ξu m dτd 1 ξu m 1 Dξũ 1 m dτ Dξu 1 m 1 dτ DξU 1 m Dξu 1 m 1 D 1 ξũ m dτ, 1 + D 1 ξu m dτd 1 ξu m 1 ; I 2 = H m c v (η m, γ m γ m + H m k Dξu 1 m 1 dτ + η m 1 γ m [c vη ( η m, γ m H m + c vγ ( η m, γ m Γ m ]

32 164 E. Zadrzyńska and W. M. Zaj aczkowski + f 1 (1 + + f 11 (1 + Dξu 1 m 1 dτ p γ (η m 1, γ m 1 Dξu 1 m 1 Γ m Dξu 1 m 1 dτ (3.1 [con.] γ m [p γη ( η m, γ m H m + p γγ ( η m, γ m Γ m ]D 1 ξu m 1 + f 12 (1 + + f 13 (1 + G 1 = f 14 (1 + + f 15 (1 + ( + f 16 (1 + G 2 = f 17 (1 + Dξu 1 m 1 dτ γ m p γ (η m, γ m DξU 1 m Dξũ 1 m dτ γ m p γ (η m, γ m D 1 ξu m 1 dτ, 1 + D 1 ξu m dτ D 1 ξu m D 1 ξu m dτ, 1 + 2(D DξU 1 1 m dτ ξ u m 2 + f 18 (1 + Dξũ 1 m dτ Dξũ 1 m dτ p(η m, γ m Dξu 1 m dτ Dξũ 1 m dτ Dξũ 1 m dτ D 1 ξu m dτ ; [p η ( η m, γ m H m + p γ ( η m, γ m Γ m ]1 ; J = f 19 (1 + D 1 ξu m dτ, 1 + D 1 ξu m dτd 1 ξγ m. D 1 ξu m dτ D 1 ξu m 1 dτ, 1 + D 1 ξu m dτd 1 ξu m ; Dξũ 1 m dτ

33 Moion of a hea conducing fluid 165 Here f i (i = 1,..., 19 are cerain smooh vecor-valued funcions of heir argumens, ũ m = θ 1 u m + (1 θ 1 u m 1, η m = θ 2 η m + (1 θ 2 η m 1, γ m = θ 3 γ m + (1 θ 3 γ m 1, < θ i < 1 (i = 1, 2, 3 and he producs of vecors in (3.1 are undersood in he sense of (2.2. In order o esimae he funcions in (3.1 we use he Hölder and Minkowski inequaliies, Lemma 2.1, Lemma 3.3, (3.91 and (3.92. ince he esimaes of F i 2,Ω, G i 3 1/2, + Dξ, 2 G i 1/4,, I i 2,Ω (where i = 1, 2 and J 3 1/2, + Dξ, 2 J 1/4, are he same as hose of F i 2,Ω, G i 3 1/2, + D2 ξ, G i 1/4,, I i 2,Ω (where i = 1, 2 and J 3 1/2, + Dξ, 2 J 1/4, we ge F 1 2,Ω c 1 (, K 1/2 Y m (, F 2 2,Ω c 2 (K sup H m 3,,Ω τ ( c 3 (, K 1/2 (Y m ( + sup H m 3,,Ω, τ I 1 2,Ω c 4 (, K 1/2 Y m (, I 2 2,Ω c 5 (K (sup τ + c 6 (, K 1/2 Y m ( + c 7 (K 1/2 (sup τ H m,ω + sup Γ m,ω τ H m 3,,Ω + sup Γ m 3,,Ω, τ G 1 3 1/2, + D 2 ξ,g 1 1/4, c 8 (, K 1/4 Y m (, G 2 3 1/2, + D 2 ξ,g 2 1/4, J 3 1/2, + D 2 ξ,j 1/4, c 1 (, K 1/4 Y m (, c 9 (, K 1/4 (Y m ( + sup H m 3,,Ω, τ where Y m ( = U m 4,Ω + sup τ U m 3,,Ω + Γ m 4,Ω + sup τ Γ m 3,,Ω and c i (i = 1,..., 1 are nonnegaive increasing coninuous funcions of heir argumens. Now inequaliies (3.99 (3.11 yield (3.12 Y m+1 ( φ(, K U m+1 2,Ω + φ 1 (, K 1/4 Y m ( By Lemma 3.3, + φ 2 (, K (sup τ (3.13 sup τ H m 3,,Ω + sup τ H m,ω c sup H m 3,,Ω τ H m,ω + sup Γ m,ω. τ

34 166 E. Zadrzyńska and W. M. Zaj aczkowski and since Γ m (τ = τ sγ m (s ds we have (3.14 sup Γ m,ω c 1/2 Γ m 4,Ω. τ Nex, consider problem (3.98. Inegraing ( wih respec o ime we obain H m (ξ, = exp ( div um u m dτ [ ( η m 1 (div um u m div um 1 u m 1 exp herefore, we have (see [15] (3.15 sup H m 3,,Ω φ 3 (, K 1/2 U m 4,Ω. τ div um u m d ] d. aking ino accoun (3.12 (3.15 and he Gronwall inequaliy we ge (3.16 Y m+1 ( φ 4 (, K 1/4 Y m (. Hence by (3.16 and (3.15 for (where is sufficienly small he sequence {(u m, γ m, η m } converges o a limi (u, γ, η W 4,2 2 (Ω W 4,2 2 (Ω C (, ; Γ 3,3/2 (Ω which is a soluion of (3.1. Moreover, from Lemma 3.3 i follows ha η L 2 (, ; W2 3 (Ω and η L 2 (, ; W2 1 (Ω. Uniqueness of (u, γ, η can be proved in he sandard way. his complees he proof of he heorem. 4. Local exisence in he case σ. In he case σ problem (1.1 in he Lagrangian coordinaes akes he form ηu µ 2 uu ν u u u + u p(η, γ = ηg in Ω, η + η u u = in Ω, ηc v (η, γγ κ 2 uγ = γp γ (η, γ u u (4.1 + µ 3 (ξ xi ξ u j + ξ xj ξ u i 2 + (ν µ( u u 2 + ηk in Ω, 2 i,j=1 u (u, p n σ (X u (ξ, = p n on, n u γ = γ on, u = = v, η = = ϱ, γ = = ϑ in Ω.

35 Moion of a hea conducing fluid 167 he argumen used o prove local exisence of problem (4.1 is he same as in he case σ = (see ec. 3. We consider he problem (4.2 ηu µ 2 wu ν w w u = F in Ω, Π ΠD w (u n = G 1 on, n D w (u n σn ( = G 2 + σ u(τ dτ H(τ dτ on, u = = u in Ω, where Π and Π are he projecions defined by Π g = g (g n n and Πg = g (g nn, respecively. he following heorem holds (see [15]. heorem 4.1. Le W 4+1/2 2, F W 2,1 2 (Ω, G i W 3 1/2,3/2 1/4 2 (, Dξ, 2 G i 1/4, < (i = 1, 2, Dξ αg 1( =, Dξ α(n D ξ (u( n G 2 ( = ( α 1, u W2 3 (Ω, H W 2 1/2,1 1/4 2 (, η L (, ; Γ 2,1 (Ω C α (Ω (α (, 1, 1/η L (Ω, w W 4,2 2 (Ω C (, ; Γ 3,3/2 (Ω, <. Assume ha is so small ha (4.3 a ( w 4,Ω + w( 3,,Ω ( φ w ξ,ω d, 1/η,Ω, η,ω, w 4,Ω, w( 3,,Ω, δ, where φ is a nonnegaive nondecreasing coninuous funcion of is argumens, a > is a consan and δ > is sufficienly small. hen here exiss a unique soluion u W 4,2 2 (Ω C (, ; Γ 3,3/2 (Ω o problem (4.2 saisfying he esimae [ 2 (4.4 u 4,Ω φ 1 ( 1/η,Ω, η,ω, F 2,Ω + G i 3 1/2, + 2 i=1 D 2 ξ,g i 1/4, + H 2 1/2, + φ 2 ( η 2,,,Ω, η C α (Ω u 2,Ω ] + φ 3 ( 1/η,Ω, u( 3,,Ω, i=1

36 168 E. Zadrzyńska and W. M. Zaj aczkowski where φ i (i = 1, 2, 3 are nonnegaive increasing coninuous funcions of heir argumens. Now consider he problems η m u m+1 µ 2 u m u m+1 ν um um u m+1 = um p(η m, γ m + η m g in Ω, (4.5 Π Π um D um (u m+1 n(u m = on, n D um (u m+1 n(u m σn m ( u m+1 (τ dτ = n n(u m [p(η m, γ m p ] + σn m (ξ on, u m+1 = = v in Ω ; η m c v (η m, γ m γ m+1 κ 2 u m γ m+1 = γ m p γ (η m, γ m um u m (4.6 + µ 3 (ξ xi ξ u mj + ξ xj ξ u mi 2 2 i,j=1 (4.7 + (ν µ( um u m 2 + η m k in Ω, n(u m um γ m+1 = γ on, γ m+1 = = ϑ in Ω ; η m + η m um u m = in Ω, η m = = ϱ in Ω. For he zero sep funcions u, γ and η we ake he funcions given by formulae (3.74 (3.8. Using heorems 4.1 and 3.6 o problems (4.5 and (4.6, respecively and repeaing he argumen from he proof of heorem 3.7 we obain heorem 4.2. Le W 4+1/2 2, f C 2,1 (R 3 [, ], r C 2,1 (R 3 [, ], ϑ C 2,1 (R 3 [, ], v W 3 2 (Ω, ϑ W 3 2 (Ω, 1/ϑ L (Ω, ϑ >, ϱ W 3 2 (Ω, 1/ϱ L (Ω, ϱ >, c v C 2 (R 2 +, c v >, p C 3 (R 2 +. Moreover, assume ha he following compaibiliy condiions are saisfied: (4.8 D α ξ [D ξ (v n (p (ϱ, ϑ p n σ (ξ] =, α 1, on,

37 and Moion of a hea conducing fluid 169 (4.9 D α ξ (n ξ ϑ ϑ(ξ, =, α 1, on. Le be so small ha < c 1 (1 CK 3 de{x ξ } c 2 (1 + CK 3 (where x(ξ, = ξ + u (ξ, τ dτ for, u is given by (3.74, K is given by (3.91 and he consan C = C(K which is a nondecreasing coninuous funcion of K is given by (3.94. hen here exiss, < such ha for here exiss a unique soluion (u, γ, η W 4,2 2 (Ω W 4,2 2 (Ω C (, ; Γ 3,3/2 (Ω of problem (4.1 and sup u 4,Ω + γ 4,Ω CK, η 3,Ω + sup η 2,Ω + η L2 (, ;W2 3 (Ω + η L2 (, ;W2 1(Ω Φ 1 (, a K ϱ 3,Ω, 1/η,Ω + η,ω Φ 2 ( 1/2 K 1/ϱ,Ω + Φ 3 ( 1/2 K ϱ,ω, where Φ i (i = 1, 2, 3 are increasing coninuous funcions of heir argumens, a >. References [1] R. A. Adams, obolev paces, Academic Press, [2] G. A l l a i n, mall-ime exisence for he Navier okes equaions wih a free surface, Appl. Mah. Opim. 16 (1987, [3] O. V. Besov, V. P. Il in and. M. Nikol skiĭ, Inegral Represenaions of Funcions and Imbedding heorems, Nauka, Moscow, 1975 (in Russian. [4] L. Landau and E. Lifschiz, Hydrodynamics, Nauka, Moscow, 1986 (in Russian. [5]. N i s h i d a, Equaions of fluid dynamics: free surface problems, Comm. Pure Appl. Mah. 39 (1986, [6] P. ecchi and A. Valli, A free boundary problem for compressible viscous fluids, J. Reine Angew. Mah. 341 (1983, [7] V. A. o l o n n i k o v, A priori esimaes for parabolic equaions of second order, rudy Ma. Ins. eklov. 7 (1964, (in Russian. [8], On boundary problems for linear parabolic sysems of differenial equaions of general ype, rudy Ma. Ins. eklov. 83 (1965 (in Russian; English ransl.: Proc. eklov Ins. Mah. 83 (1967. [9], On he solvabiliy of he iniial-boundary value problem for equaions of moion of he viscous compressible fluid, Zap. Nauchn. em. LOMI 56 (1976, (in Russian. [1], On an unseady moion of an isolaed volume of a viscous incompressible fluid, Izv. Akad. Nauk R er. Ma. 51 (5 (1987, (in Russian. [11], On an iniial-boundary value problem for he okes sysem which appears in free boundary problems, rudy Ma. Ins. eklov. 188 (199, (in Russian. [12] V. olonnikov and A. ani, Free boundary problem for a viscous compressible flow wih surface ension, in: Consanine Carahéodory: An Inernaional ribue,. M. Rassias (ed., World cienific, 1991,

38 17 E. Zadrzyńska and W. M. Zaj aczkowski [13] W. M. Zaj aczkowski, On local moion of a compressible baroropic viscous fluid bounded by a free surface, in: Parial Differenial Equaions, Banach Cener Publ. 27, Ins. Mah., Polish Acad. ci., Warszawa, 1992, [14], On nonsaionary moion of a compressible baroropic viscous fluid bounded by a free surface, Disseraiones Mah. 324 (1993. [15], On nonsaionary moion of a compressible baroropic viscous capillary fluid bounded by a free surface, o appear. [16], Local exisence of soluions for free boundary problems for viscous fluids, o appear. INIUE OF MAHEMAIC INIUE OF MAHEMAIC ECHNICAL UNIVERIY OF LÓDŹ POLIH ACADEMY OF CIENCE AL. POLIECHNIKI 11 ŚNIADECKICH LÓDŹ, POLAND -95 WARAW, POLAND Reçu par la Rédacion le